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Math Help - complex number question

  1. #1
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    complex number question

    1) Use De Moivre’s theorem to show that (1 + i \tan(x))^5 = \frac{1}{\cos^5(x)} cis (5x)

    2) Hence, find expressions for cos (5x) and sin (5x) in terms of tan(x) and cos(x).
    Last edited by scorpion007; October 25th 2006 at 11:20 PM. Reason: made it more clear
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  2. #2
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    ah i got the first one... i think.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by scorpion007 View Post
    1) Use De Moivre’s theorem to show that (1 + i \tan(x))^5 = \frac{1}{\cos^5(x)} \cos (5x)
    This should be:

    <br />
(1 + i \tan(x))^5 = \frac{1}{\cos^5(x)} [\cos (5x) + i \sin(5x)]<br />

    2) Hence, find expressions for cos (5x) and sin (5x) in terms of tan(x) and cos(x).
    As you have done part (1) we need only worry about part (2)

    (1 + i \tan(x))^5 = \frac{1}{\cos^5(x)} (\cos (5x) + i \sin(5x))

    implies:

    \cos^5(x) (1 + i \tan(x))^5 = \cos (5x) + i \sin(5x)

    so expanding the second term on the left and equating real and imaginary
    parts we will have the answer required - but I have to rush off now, may be
    able to finish this later.

    RonL
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  4. #4
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    hi, CaptainBlack, thanks. For the actual equation i wrote "cis" which is not a typo, as you guessed "cis(x)" means "cos(x + i sin(x)". It is just an abbreviated form we commonly use. Maybe you don't use it where you are.

    Now for the second part, is there a fairly simple way to expand (1 + i tan(x))^5 ? Expanding a degree 5 poly is a bit of a pain isn't it?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by scorpion007 View Post
    hi, CaptainBlack, thanks. For the actual equation i wrote "cis" which is not a typo, as you guessed "cis(x)" means "cos(x + i sin(x)". It is just an abbreviated form we commonly use. Maybe you don't use it where you are.

    Now for the second part, is there a fairly simple way to expand (1 + i tan(x))^5 ? Expanding a degree 5 poly is a bit of a pain isn't it?
    (1 + i tan(x))^5 = 1 + 5 (i tan(x)) + 10 (i tan(x))^2 + 10(i tan(x))^3 +

    .........................5(i tan(x))^4 + (i tan(x))^5

    .......................= 1 + 5 i tan(x) - 10 (tan(x))^2 - 10 i (tan(x))^3 +

    .........................5(tan(x))^4 + i (tan(x))^5

    .......................= [1 - 10 (tan(x))^2 + 5(tan(x))^4] +

    ..........................i [5 tan(x) - 10 (tan(x))^3 + (tan(x))^5]

    RonL
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